Seminar on Singularities of Solutions of Linear Partial Differential Equations. (AM-91), Volume 91 9781400881581

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Annals of Mathematics Studies Number 91

SEMINAR ON SINGULARITIES OF SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS E D IT E D B Y

LARS HORMANDER

P R IN C ET O N U N IV E R S IT Y PR ESS AND U N IV E R S IT Y O F TO K Y O PRESS

PR IN C ETO N , N EW JE R S E Y 1979

Copyright © 1979 by Princeton University Press ALL RIGHTS RESERVED

Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

CONTENTS PREFA CE

vii

SPECTRA L ANALYSIS OF SINGULARITIES by L. Hormander 1. Introduction 2. Definition and b asic properties of the singular spectrum 3. The non-characteristic regularity theorem 4. Pseudo-differential operators 5. B ich aracteristics and symplectic geometry 6. Fourier integral operators corresponding to canonical transformations 7. Further equivalence theorems 8. Propagation of singularities and semi-global existen ce theorems for pseudo-differential operators satisfying condition (P ) FOURIER INTEGRAL OPERATORS WITH COMPLEX PHASE FUNCTIONS by J . Sjostrand 0. Introduction 1. L ocal study 2. Lagrangean manifolds associated to phase functions 3. Global definition of Fourier integral distributions 4. Fourier integral operators 5. Application to the exponential of a pseudo-differential operator H YPO ELLIPTIC OPERATORS WITH DOUBLE CHARACTERISTICS by A. Menikoff 1. Conditions for hypoellipticity 2. The asymptotic behavior of the eigenvalues D IFFEREN TIA L BOUNDARY VALUE PROBLEMS OF PRINCIPAL T Y P E by R. B. Melrose 1. Introduction 2. Examples 3. Symplectic geometry 4. Pseudo-differential operators 5. Fourier integral operators 6. Normal forms 7. Other boundary conditions 8. Higher-order tangency 9. Example (2.3) again v

3 3 12 16 22 27 34

39

51 51 53 55 57 58

65 71

81 83 84 90 93 96 100 101 106

vi

CONTENTS

PROPAGATION OF SINGULARITIES FOR A CLASS OF OPERATORS WITH DOUBLE CHARACTERISTICS by N. Hanges 0. Introduction 113 1. Statement of results 113 2. Reduction to canonical form 116 3. A simple example 117 4. R esults independent of the lower order terms 118 5. Results depending on the lower order terms 121 SU BELLIPTIC OPERATORS by L . Hormander 1. Introduction 2. The Taylor expansion of the principal symbol 3. N ecessary conditions for subellipticity 4. L ocal properties of the principal symbol 5. Estim ates for the localized operators 6 . Proof of the sufficiency in Theorem 3.4 7. Calculus lemmas ' 8 . Concluding remarks LACUNAS AND TRANSMISSIONS by L . Boutet de Monvel 1. Sharp fronts 2. Transmissions 3. Boundary value problems 4. Symmetry of the elementary solution of a hyperbolic equation 5. Further developments

127 129 141 149 167 188 200 207 209 212 213 215 218

SOME CLASSICAL THEOREMS IN SPECTRAL THEORY REVISITED by V. Guillemin 0. Introduction 1. The lattice point problem 2. Weyl-type formulas 3. Szego-type formulas I 4. Szego-type formulas II

219 221 228 238 247

A SZEGO THEOREM AND COMPLETE SYMBOLIC CALCULUS FOR PSEUDO-DIFFERENTIAL OPERATORS by H. Widom 1. Introduction 2. Pseudo-differential families in R n 3. The half-space problem 4. Pseudo-differential operators on manifolds 5. The heat expansion 6 . Functional calculus

261 264 267 272 280 281

PREFA CE Apart from some modifications and additions this book co n sists of the notes of a seminar held at The Institute for Advanced Study in 1977-78. Singularities of solutions of differential equations form a common theme. Some of the lectures were devoted to the analysis of singularities and some to applications in spectral theory. In an introductory series of lectures the basic techniques of pseudo­ differential and Fourier integral operators were developed from this point of view and the basic facts concerning singularities of solutions of opera­ tors of principal type were outlined.

These lectures form the main part of

the first paper here but in the last section some new results on the propa­ gation of singularities (and dual existen ce theorems) have been added al­ though they were presented later on in the seminar.

For operators which

are not of principal type but can be written modulo terms of lower order as a product of two such factors, the propagation of singularities is studied in the paper by N. Hanges. A pseudo-differential operator P

is called hypoelliptic if a distribu­

tion u can only be singular where Pu is.

Two c la s s e s of hypoelliptic

operators are discussed here. We give a self-contained presentation of Egorov’s characterization of subelliptic operators, and the first part of the paper by A. Menikoff is devoted to hypoellipticity of operators with double ch aracteristics.

(c) 1978 Princeton U niversity P ress Sem inar on Singularities of Solutions 0 -6 9 1 -0 8 2 2 1 -9 /7 8 /0 v ii $ 0 0 .5 0 /1 (cloth) 0 -6 9 1 -0 8 2 1 3 -8 /78/O v ii $ 0 0 .5 0 /1 (paperback) Fo r copying information, see copyright page

vii

viii

PREFACE

In the papers mentioned so far the main technical tools have been pro­ vided by pseudodifferential operators and the simplest types of Fourier in­ tegral operators. However, these are often insufficient to describe the singularities of important objects such as fundamental solutions.

Fourier

integral operators based on complex phase functions have a much wider scope.

They were presented by J . Sjostrand in two introductory lectures

contained in his paper here. The study of boundary problems requires further development of pseudo- and Fourier integral operators.

In his paper

R. Melrose indicates how one can define these concepts in a manifold with boundary so that the desirable properties are obtained.

Applications of

these new techniques are outlined in the study of propagation of singulari­ ties for mixed boundary problems in the ca s e of non-tangential, diffractive or gliding rays.

New results by Melrose and Sjostrand on propagation of

singularities along rays which are tangent to the boundary of finite but arbitrarily high order are given at the end of his paper. Description of a fundamental solution, say, as a Fourier integral distri­ bution gives a characterization of its singularities. However, it is often of interest to be able to conclude regularity properties of a more cla s s ic a l kind such as smoothness on one side of a hypersurface and existen ce of a smooth extension across it. The study of lacunas is devoted to such prop­ erties, and the paper by L . Boutet de Monvel here sums up some recent work in the area by L . Garding, A. Hirschowitz and A. Piriou. Since the pioneering work by Carleman it is well known that there is an intimate relation between the asymptotic properties of eigenfunctions (eigenvalues) of differential operators and the singularities of various transforms of them such as the resolvent, Laplace or Fourier transform. The second part of the paper by Menikoff and that of V. Guillemin are de­ voted to such applications in spectral theory. For hypoelliptic operators P with double ch aracteristics Menikoff presents joint work with Sjostrand on the singularities of e_tP erties of the eigenvalues.

as t -> 0 , and the resulting asymptotic prop­

The first part of the paper by V. Guillemin is

ix

PREFACE

devoted to the remainder term in the asymptotic formulas for the eigen­ values of elliptic operators and the number of lattice points in a large ball. The second part starts from the cla s s ic a l theorem of Szego on the asymptotic properties of the eigenvalues of the projection of a multiplica­ tion operator on sp aces of trigonometric polynomials of increasing order. Guillemin substitutes the range of the spectral projections of an elliptic operator for the space of trigonometric polynomials of given degree and obtains similar results with a number of applications.

Finally the paper

by H. Widom is devoted to the study of higher order terms in the Szego theorem and the pseudo-differential operator technique which this requires. This book makes no claim to cover completely the recent work on the singularities of solutions of linear partial differential equations. In par­ ticular the analytic (hyperfunction) theory is only mentioned briefly for it was the subject of a parallel seminar. Nevertheless it is hoped that this volume can serve as an introduction to the subject in the absence of a more system atic treatment. Lars Hormander THE IN STITUTE FO R ADVANCED STUDY

Seminar on Singularities of Solutions of Linear Partial Differential Equations

SPECTRA L ANALYSIS OF SINGULARITIES Lars Hormander1 ^ 1.

Introduction The existen ce of solutions of a linear partial differential equation is

closely related to the singularities which solutions of the adjoint equation can have. We shall therefore study singularities of solutions first and d iscu ss existen ce theorems afterwards as applications. By the support supp u of a function or distribution u one means the sm allest closed se t such that u vanishes in the complement. Similarly the singular support sing supp u is the sm allest closed set such that u is a C°° function in the complement. However, it is possible to make a harmonic analysis of u at the singularities which describes them much more precisely and at the same time simplifies the study of the singulari­ ties of solutions of partial differential equations. We shall introduce this harmonic decomposition of the singularities in the next section and then proceed to develop the main tools for studying it in the context of differen­ tial equations of principal type where it originated.

2.

Definition and basic properties of the singular spectrum If v e g (R n) is a distribution with compact support, the Fourier trans­

form v is defined by ^Supported in part by NSF grant M CS77-18723.

© 1978 Princeton University P ress Sem inar on Singularities of Solutions 0 -6 9 1 -0 8 2 2 1 -9 /7 8 /0 0 0 3 $ 0 2 .3 5 /1 (cloth) 0 -6 9 1 -0 8 2 1 3 -8 /7 8 /0 0 0 3 $ 0 2 .3 5 /1 (paperback) For copying information, see copyright page

3

4

LARS HORMANDER

v (f) = v (e -J< •’ £> ), f f R n . It is well known that v e C°° if and only if Iv O

(2 .1 )

s CN( l+ | ^ | r N, N = 1, 2, •••; £ t R n .

If v is not in C°° we can describe the spectrum of the singularities by introducing the cone S(v) C R n \ 0 consisting of directions where (2.1) is not valid.

Thus

77

t 2 (v ) means that there is an open cone V ?

such

77

that (2.1) is valid when £ e V . Clearly 2 (v ) is a closed cone in Rn \ 0 , and

2

(v) =

0

if and only if v e C^3.

Whilesing supp v

only describes the location ofthe singularities,

the set 2 (v ) only describes the frequencies causing them. To combine the two types of information we need

If e (Rn) and v < r 6 ( R n) ,

L e m m a 2.1 .

( 2 .2 )

2 (< £ v )

C

then

X (v ) .

Proof. The Fourier transform of u = c£v is (2n)~n4> * v , where rapidly decreasing and Iv(£)| ^ C (l+

|)N for some N .

is

IfT is an

open cone where v is rapidly decreasing we write

|u(

0

when

77

e C r,

so the integral can be estimated by

some

S PE C T R A L ANALYSIS OF SINGULARITIES

J

l*\€\ which is rapidly decreasing as £ -> . This proves the lemma. If u * $ ( 0 ) where 0

\ =

(2 .3 )

f]

is an open set in R n , we set for x e H

2

(1 ,

€ C^(fl) with i

(x) ••• 0|. If V is an open 77

then

is true when | f | + t = l .

- tr?| > c(| 0 ,

for this

Hence | £-k 2 77)/k > c(|f |+ k2)/k > cy/\^\ . It

follows that G satisfies ( 2 . 1 ) in ()v which proves the assertion. Starting from this example it is easy to show that there is a continuous function u with sing spec u equal to a given conic set.

In fact, after

suitable topologies have been introduced as we shall do for distributions below, this is a standard condensation of singularities argument. We shall now discuss how sing spec u transforms if u is composed with a C°° map \[i. It is well known that the composition is well defined when the Jacobian matrix if/' is surjective.

However, the notion of singu­

lar spectrum allows us to define the composition more generally. we shall first introduce a topology in

To do so

7

SPECTRAL ANALYSIS OF SINGULARITIES

JTp(Q) = |ue ® ( f i) , sing spec u C T| where T is a closed cone Cfi x (R n \Q). R ecall that u e 5 )p

means

precisely that for every e Cq (Q) and every closed cone V C R n with (supp cf> x

(2 . 5 )

v) n r

=

0

we have (2 .6 )

sup |£|n |Si(£)| < « ,

N= 1 , 2 , -

.

V

For a sequence Uj e C ^ (fl), V is a closed cone and (2 .5 ) is valid. This implies that u e ® p (^ ) . Since (i) implies that u uniformly on every compact set and N is arbitrary in (ii), we can replace (ii) by (ii)'

sup sup |£|n | 0 in C ^(R n) with

dx = 1 and

support shrinking to 0 so fast that supp y j + supp i/fj C 1) . Then

uj = y j * OAju) . C ~ (0 ) , and Uj -> u in 2 )p (fi).

In fact, if

e C^(H) we have for large j

8

LARS HORMANDER

If V is a closed cone and (2.5) is valid then we can choose i/r e C^(Q) equal to 1 in a neighborhood of supp 0

and a closed cone W with

interior containing V \ 0 so that ((supp i/r) x W) D F = 0 . Hence = 0 ( y j * OAu)) = 0Wj for large j , where

IWj| = Ivjl Since |^ru |

\ $ u \


x' e R n _ 1 ,

so sing spec u C |(x', x n, £ ', 0); ( x ',g ') e sing spec v| . Here equality must hold sin ce v is the pullback of u with the map x'-> (x', xn) for any fixed xn . It follows that sing spec u can be any closed conic subset of the ch aracteristic set

=

0

which is invariant

under translations in the x n direction, and no other se ts can occur. Our aim is to establish a general form of this result. However, before doing so it is useful to take another look at the proof of Theorem 3.1.

The

main point there was that the equation *P (x,D )v = f ) /a !

where again we can introduce the asymptotic series for a and for b . (For a detailed proof along these lines see [ 6 , p. 143-148].) If a ~ ^ and b ~

> b,

• then the highest order term in c

am-j

is a mb . Given a

and a non-characteristic point (x 0 , £ 0), £ Q ^

0

, that is,

a m(x0 ,

^ ^'

it is always possible to find b of order -m

so that the symbol of the

composition minus 1 is rapidly decreasing in a conic neighborhood V of (x 0 , D) starts with b_mam 0

so we first choose b_m so that b-m m

m

which can be done by multiplication of

in a m_ 1

V by a suitable cutoff function

on ft x Sn _ 1 . The term in b(x,D ) a(x, D) of order - j

is a sum of

b_m_j am and a homogeneous function q of order - j

determined by

b_m, •••,-b_m_ j + 1

which we assume already chosen. If we take b_m_j =

- q / a m in V the inductive choice continues, and the statement is proved. Theorem 3.1 can now be extended to properly supported pseudo­ differential operators: (4 .7 )

sing spec u C sing spec au U Char a .

In fact, if (xQ, £ q) \ Char a we can choose a properly supported pseudo­ differential operator b(x, D) such that b(x, D) a(x, D) = identity + c(x , D) where c is of order -oo in a conic neighborhood of (x Q, £ 0) . Then u = b(x, D) a(x, D ) u - c ( x , D)u , sing spec b(x, D) a(x, D)u C sing spec a(x, D)u by (4 .5 ), and (4 .6 ) implies that (x 0 , a a ( x ,- £ )/a ! , 2

(iDxDf>a a(x, . real valued and dcf) £ ( 4 .3 /

0

If 0

is

in supp u then

e_lt

where c has an asymptotic expansion in homogeneous functions which is of order -oo except in a conic neighborhood of (yQ, rj0) where it is the product (f 1 ° y ) f of the principal symbols of Fj factor e 771*7 ^4

and F

times possibly a

which is a constant of modulus one which we ignore since

it can be absorbed in F 1 say.

But this means that the composition is a

pseudo-differential operator with symbol c . At a non-characteristic point it preserves the singular spectrum so we conclude that (6 .8 )

(y , 77)

6

sing spec u < = > y ( y , 7/) e sing spec Fu if Fm( y , 7/) / 0 .

It is not hard to prove (see [7, section 4.2]) that the adjoint operator of F modulo an operator with C°° kernel is a Fourier integral operator belonging to y _ 1 . (Note that it follows from (6.4) that our definition of this concept is independent of the choice of local coordinates.) This im­ plies that multiplication by pseudo-differential operators to the right also preserves the c la s s of Fourier integral operators belonging to y , and it shows that operators of order 0 are continuous from L

2

S

' /

fife

2

to ^loc

SPECTRAL ANALYSIS OF SINGULARITIES

since F * F y (y , 77) if F

is.

More generally, if u e

31

at (y, 77) then Fu e

at

is of order 0 .

We have now developed the tools which allow us to complete the proof of Theorem 5.1.

We keep the notations there and let (x Q, £ 0) be a ch arac­

teristic point with (x Q, f Q) \ sing spec Pu .

(6 .9 )

We may assume that Hp does not have the radial direction at (x Q, £ 0) for there is nothing to prove then. By Theorem 5.2 we can then find a canonical transformation from a neighborhood of (0, rj0) e T *(R n) \ 0

to a

neighborhood of (xQ, 1 .

(His result is incomplete with respect to removal of

lower order terms which requires an analogue of Lemma 6.1 with two different operators A .) The differential operator D 1 + ix 1 ^D2

is a modification first con­

sidered by Mizohata [12] of the famous example of H. Lewy of a differen­ tial equation with no solutions.

P R O P O S IT IO N 7.2.

If k IS odd and r/ 0 1 = 0, r)02 < 0 then one can find

a C 1 solution of the equation (D 1 + ix i kD2 )u = 0 with sing spec u =

Uo, tn0), t> oi. Thus the solution has a singularity which does not propagate. Proof. L et v (x'), x' = (x2 , '*’ >x n) be a function of compact support with sing spec v = 1(0, ttjq'), t > 01.

(See Example 2 .3 .) Then v is rapidly

decreasing outside any conic neighborhood V of rjQ'.

exp ( f 2 x x k+1 /(k + 1 ) + i < x ',£ '>)v(£')d u e H^+m

for every t < s .

From (8.1) we can obtain existen ce theorems exactly as in section after Theorem 6.3.

6

In particular, the equation ^Pu = f can be solved in a

neighborhood of K for all distributions f which are orthogonal to a finite dimensional subspace of C q (K );

one can find u with m-1 derivatives

more than f . This extends the results of Nirenberg and Treves [13] as well as those of B eals and Fefferman [1], Theorem 8.2 was proved by Duistermaat and Hormander [2] under the assumption that at every ch aracteristic point the Hamilton fields of Re p and Im p are linearly independent of the radial vector field d /d £ . Condi­ tion (P ) implies that {R e p, Im pi = 0 on the ch aracteristic set so it must then be an involutive manifold of codimension 2 .

Thus it has a natural

foliation by two dimensional leaves B with tangent plane spanned by HRe p and HIm L2

. Now recall the notation s*(x,< f) for the number of

derivatives of u at ( x , £ ) .

It was proved in [2] that if Spu > s then

min (s* , s+m - 1 ) is a superharmonic function with respect to the analytic structure defined in B by the Hamilton field H p. Now suppose that u has the properties in (8.1).

The lower semi-continuous function s * must

assume its infimum, and this must occur at a ch aracteristic point ( x , f ) if it is < s+ m -1 , for s * ^ s+m at the non-characteristic points.

But then

the superharmonic function min (s * , s+m - 1 ) must be constant in the leaf through ( x , f )

which is a contradiction with the assumption which states

that it contains points over C k There is another situation where the following improvement of Theorem 6 .2 will give Theorem 8.2:

THEOREM 8.3.

L et P be a pseudo-differential operator of order 1 with

principal symbol p , and let RDI ? t -> y(t) e T * ( f 2 ) \ 0

be a finite inter­

val on a bicharacteristic for Re p . Assum e that Im p ^ 0 in a neighbor­ hood of y (I ) . If u e 3) (12), Pu eat y(I) and u *

H (S )

at y(b) when

b is the right hand end point of I , then u e H(S) at y(I) .

41

SPECTRAL ANALYSIS OF SINGULARITIES

The proof was given in Hormander [10]. set of p and let

L et (2 be the ch aracteristic

be the set of points lying on some sem ibicharac­

teristic y with a non-characteristic end point.

From Condition (P ) it

follows easily that P must then satisfy the hypothesis of Theorem (8 .3 ) at y after multiplication by an elliptic factor, so ( 8 . 1 ) follows microlocally at

. Note that a sem ibicharacteristic starting at a point in

must always remain in the ch aracteristic set. The preceding arguments must be extended to give a general proof of Theorem 8 .2 .

An obvious difficulty is of course that as noted already we

sometimes have propagation of singularities along curves and sometimes propagation along two dimensional manifolds. can be nicely separated. L et us denote by C 2

Fortunately these two ca s e s the set of ch aracteristic

points where d Re p and d Im p are linearly independent.

As already

observed this is an involutive manifold of codimension 2 . Now we extend C 2 to the set C 2e of all points which can be joined by a sem ibicharac­ teristic to some point in C 2 . Note that a sem ibicharacteristic through a point in C \ (C 11 U &2 ) must be a one dimensional bicharacteristic, that is, a curve such that Hp is always proportional to the tangent vector along it.

PROPOSITION 8 .4 .

Condition ( P ) implies that

e2e

is an involutive mani­

fold of codim ension 2 .

To see this we let V be a small neighborhood of a point in C 2 . Then V fl C 2 divides the hypersurface of zeros of Re qp in V into two parts where Im qp > 0 and Im qp > 0 respectively.

The flowout along HR 0

of these se ts must lie in the sets where Im qp ^ 0 and Im qp ^ 0 respec­ tively, so the flowout of V H shows that C 2e n

= 0 .)

remains in C. and is involutive. (This It is not hard to complete the proof of Propo­

sition 8 .4 . As in the non-degenerate c a s e (?2e has a natural foliation by two dimensional leaves, which we call two dimensional b ich aracteristics. From

42

LARS HORMANDER

a point in C ^ \ C 2 we can always find a sem ibicharacteristic y to a point in C 2 and it must be a one dimensional bicharacteristic until it hits the boundary of C 2 . It is then a routine matter to show that one can transform P by conjugation with Fourier integral operators as in section so that y becomes I x £ ° = (0 ,

0

x

0

, where

I is an interval on the

+ if(x, f '= (f 2 , •••>£„) ■

Since d Im p ^ 0 at some point on I it follows from condition f(x,< f) = g ( x ',D h (x , 0

x ^ axis,

, 1 ) and p(x, BQ is the projection.

the largest function on B Q such that

7r*su

< s * , that is,

L et s u be s u is the

minimum of s * on each equivalence cla ss. If s is superharmonic in co C Bq it follows then that min (s u, s+m - 1 ) is s u p e rh a rm o n ic in co .

In the proof of this theorem the following characterization of spaces is convenient:

LEMMA 8 .7.

L et y and 0

be functions in C ^ (T *R n \ 0) and set

45

S P E C T R A L ANALYSIS OF SINGULARITIES

qt ( x , f ) = v O ^ / t ) ^ ’ ^

// f

S (Rn) and s| > Re

6

as t

•

in supp y , then ||qt(x, D)f||

0

2

is hounded

oo . Conversely, if ||qt(x, D) f|| is hounded as t -> oo then s j ( x , f ) > Re 0 ( x , f )

if y ( x , f ) ^ 0 .

Let I be a complete embedded one dimensional bicharacteristic.

An

easy modification of the proofs of Nirenberg and Treves shows that, if m=

1

,

(8.4)

||u||

C(||Pu|| + I M ) , u f C~

1

where R is a pseudo-differential operator which is of order - 1 conic neighborhood V of I . L et B be the leaf through I and

in a oj

an

open neighborhood of I in B Q such that the inverse image does not meet V . L et K be a compact set in co and h a harmonic polynomial such that

h(w) < mi n ( s n, u s ) on

where w is the local analytic coordinate.

K

The theorem will be proved if

we show that the same inequality is valid in K.

That h(w) < s follows

since s is superharmonic so what must be proved is that h(w) < s u . Now w can be considered as a solution of Hpw = 0 , and if h = Re F F

is an analytic polynomial then F(w) satisfies the same equation.

can be extended to a function

6

C~ which is equal to

1

> so that for a suitable

near K'

s * > Re cf> in supp d y ,

S£ > Re c/> in supp y .

An iterative argument allows one to assume also that s* > Re cf> - 8 supp y

It

satisfying the equation in a neighbor­

hood of the inverse image K' of K , in y

where

for some fixed but arbitrarily small 8 > 0 .

in

Now we apply (8 .4 )

with u replaced by q^.(x, D)u where q^. is defined in Lemma 8 .7 .

The

46

LARS HORMANDER

main difficulty is then to estimate ||Pqtu|| or just ||[P, qt]u|| . The lead­ ing term becomes -i(H py ( x , e / t ) + v ( x , f / t ) l o g t

.

If Hp were 0 , which one can achieve by the Cauchy-Kovalevsky theorem in the analytic ca s e , then the proof would be finished. In general one can only make sure that Hp - C ||u||( s ) 2 , u ( C ” , provided that 2s ^ m - 6 ( p - 8 ) / S . In particular one can take s = 0, m = 1 , p = 1-5

if 8
0 is not

identically 0 there. Then + i h ( x ,£ ') (g ( x '

, 0

+ £ n+1)

47

S P E C T R A L ANALYSIS OF SINGULARITIES

also satisfies condition (P ) and there is now a two dimensional bicharac­ teristic through I x

. A solution of Pu = f in R n can be considered

as a distribution in R n+ 1 8.6

which is independent of x n+ 1

and Theorem

implies an analogous result on the regularity on I where superharmo-

nicity is replaced by concavity with respect to the parameter /h ( x 1 , 0 , f /O) dx1 on I . (Intervals where this is constant should be identified to a point.)

LEMMA 8 . 8 . Suppose that ^

I x

0

+ if(x,

0

0

and then r must still vanish when

).

The proof follows from Malgrange’s preparation theorem. F . Treves has given an example which shows that r cannot always be taken equal to

0

.

When r = 0 we have just seen that Theorem

8.6

leads to a concavity

statement for s * on I . When r is not identically 0 ,

the fact that r

vanishes of infinite order on the se t of sign changes combined with the im­ provement of Theorem 8 .3 mentioned above, which permits one to get good control away from the set of sign changes, also leads to a slightly weaker result on the propagation of singularities along I . L et us now return to the proof of (8.1).

The results on propagation of

singularities discussed above show that if u satisfie s the hypothesis in ( 8 . 1 ) and inf s * < s+m - 1 , then the infimum cannot be attained at points of (?n , C2e or the set

£ 12

C C \ ( £ u U (?2e) consisting of bicharacter-

48

LARS HORMANDER

istics of the form discussed in Lemma

8 .8

. What remains is the set

£3

consisting of b ich aracteristics where p can be put in the form £ ^ + if(x, with f vanishing to the third order. Now it turns out that the proofs of B eals and Fefferman give global show that the infimum of s *

estim ates near this set, and they

cannot be assumed in (23 only.

The con­

clusion is that s* > s+m-1 which completes the proof. Detailed proofs a re given in a paper to appear in Annals of Mathematics. R EFER EN C ES [1]

R. B eals and C. Fefferman, On local solvability of linear partial differential equations. Ann. of Math. 97(1973), 482-498.

[2]

J . J . Duistermaat and L . Hormander, Fourier integral operators II. Acta Math. 128(1972), 183-269.

[3]

J- J . Duistermaat and J . Sjostrand, A global construction for pseudodifferential operators with non-involutive ch aracteristics. Inv. Mat. 20(1973), 209-225.

[4]

Yu.V. Egorov, On canonical transformations of pseudo-differential operators. Uspehi Mat. Nauk 25(1969), 235-236.

[5] [6 ]

______________, Subelliptic operators. Usp. Mat. Nauk 30:2(1975), 57-114 (Russian Math. Surveys 30:2(1975), 59-118). L. Hormander, Pseudo-differential operators and hypoelliptic equa­ tions. Proc. Symp. Pure Math. 10(1966), 138-183.

[7]

—____________ , Fourier integral operators I. Acta Math. 127(1971), 79-183.

[8 ]

______________, Linear differential operators. 1970, 1, 121-133.

[9]

---------------------, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients. Comm. Pure Appl. Math. 24(1971), 671-704.

A ctes Congr. Int. Math.

[10] _____________ , On the existen ce and the regularity of solutions of linear pseudo-differential equations. Ens. Math. 17(1971), 99-163. [11] R. Melrose, Equivalence of glancing hypersurfaces. Inv. Mat. 37 (1976), 165-191. [12] S. Mizohata, Solutions nulles et solutions non analytiques. J . Math. Kyoto Univ. 1(1962), 271-302. [13] L . Nirenberg and F . Treves, On local solvability of linear partial differential equations. Part I: N ecessary conditions. Comm. Pure Appl. Math. 23(1970), 1-38. P art II: Sufficient conditions. Comm. Pure Appl. Math. 23(1970), 459-510.

)

SPE C T R A L ANALYSIS OF SINGULARITIES

49

[14] M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo­ differential equations. Lecture notes in math. 287(1973), 263-529. [15] F . Treves, A new method of proof of the subelliptic estim ates. Comm. Pure Appl. Math. 24(1971), 71-115. [16] K. Yamamoto, On the reduction of certain pseudo-differential opera­ tors with non-involutive ch aracteristics. J . Diff. Eq. 26(1977), 435-442.

FOURIER INTEGRAL OPERATORS WITH COMPLEX PHASE FUNCTIONS J. Sjostrand1^ 0

.

Introduction In his lectures L . Hormander introduced local Fourier integral opera­

tors in order to transform pseudo-differential operators. Another important application of Fourier integral operators is the construction of solutions to homogeneous pseudo-differential equations; Pu = 0 mod C°° . Such constructions sometimes also lead to parametrices (i.e. inverses modulo smoothing operators). In this context, it is often important to study Fourier integral operators globally as was done first by Hormander [4] and Duistermaat-Hormander [2].

If the leading symbol of the pseudo­

differential equation under study takes complex values it is often n eces­ sary to consider complex valued phase functions. We shall here outline the calculus in the complex ca s e .

This calculus was developed jointly

with A. Melin [10] and we refer to this paper for more details.

A parallel

theory has been developed by several Soviet mathematicians, see Maslov [ 8 ], Kucherenko [ 6 ,7 ] , Danilov-Maslov [1],

1.

Local study L et f l C R ” be open. By S™ (ftxRN) C S™0 (f lx R N) we denote the ^Supported in part by NSF grant MCS77-18723.

© 1978 Princeton University P ress Sem inar on Singularities of Solutions 0 -6 9 1 - 0 8 2 2 1 -9 /7 8 / 0 0 5 1 $ 0 0 . 7 0 / 1 (cloth) 0 - 6 9 1 - 0 8 2 1 3 - 8 /7 8 / 0 0 5 1 $ 0 0 . 7 0 /1 (paperback) For copying information, see copyright page

51

52

J. SJOSTRAND

space of classical symbols a(x,0) having an asymptotic expansion oo

(1.1)

a(x,0) ~ ^

am_j(x, 6), am_-(x,\0) = \m~i am_-(x,8), \> 0 .

j =0 Let V C O x R N be an open cone, R N = R N \{0!. We say that 0(x, 0) e C°°(V) is a non-degenerate phase function if the following four conditions hold: (1.2)

lm0

(1.3)

0(x,A (9) = A0(x,0), A. > 0

(1.4)

dcf> ^ 0 everywhere dcf> ^ 0 everywhere

(1.5) d(dcfy/dd^),

d(d 0

and d f^ O

where f(x) = 0 .

53 Then

P°° • I e ^ ^ x^dt = ------ ------- is a Fourier integral distribution. J0 (f(x) + iO) The main difficulty in order to define Fourier integral distributions globally is that it is often difficult to find suitable phase functions global­ ly.

One therefore has to work with sums of expressions like I(a, cf>). The

main problem then turns out to be to examine which phase functions give rise to the same sp aces of Fourier integral distributions microlocally.

2

. Lagrangean manifolds a sso cia ted to phase functions L et WC R n be open and let f

e

C°°(W). If W C C n is open with

W fl R n = W , then it is well known that there exists a function f C C°°(W) such that d f vanishes to infinite order on W and f |w = f . Moreover, f is unique up to a function vanishing to infinite order on W. (One way of constructing f is to write down what the Taylor series should be at the real points and then make it converge as in the c la s sica l Borel theorem.) We call f the almost analytic extension of f . There is also a natural notion of almost analytic manifolds in the complex domain. It turns out that modulo certain natural equivalence relations, many important geomet­ ric objects, associated with complex phase functions, are almost analytic. This terminology is quite heavy however, so in order to simplify this survey we shall, somewhat incorrectly, think of all functions in the real domain as being real analytic. The corresponding extensions are then germs of holomorphic functions defined in a small neighborhood of the real domain. If W is an open set in the real domain, then W will denote some sufficiently small open set in the complex domain which intersects the real domain along W . If A C W is a complex manifold, we put ar

= wn A. Now let cf> be a non-degenerate phase function defined in a conic

neighborhood V of a point (xo, 0 o) ^ x R N , and assume that (xo, 0 o) £ (C ^ )R . We extend cf) to some complex cone V and put

54

J. SJOSTRAND

( 2 . 1)

=

\(x,d)e V;'e(x,e)=Q\,

if V is small enough then

is a manifold of complex dimension n .

The map

(2.2)

> (x,0) t-> (x ,0 'x) f T*12\0

has injective differential so if we shrink V and V around (x Q, 0 Q) , the image A = A^ will be a conic n-dimensional manifold. It is easily veri­ fied that A is Lagrangean.

L et

(xQ, L et y(a)

6

0

n k (0 ,0 ) in R x R . Assum e

with equality at ( 0 , 0 ), 3 y( 0 , 0 ) =

C n be the analytic solution of

for a in a small real neighborhood of 0 .

a'y(y,a) =

0

, det a"yy( 0 , 0 ) ^

0

, y( 0 ) = 0 , defined

Then there exists a constant

C > 0 such that Im a(y(a), a) > C|lm y(a ) |2 .

This lemma shows that Im H(